Gaiotto duality for the twisted A2N −1 series

We study 4D N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 superconformal theories that arise from the compactification of 6D N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = (2, 0) theories of type A2N −1 on a Riemann surface C, in the presence of punctures twisted by a ℤ2 outer automorphism. We describe how to do a complete classification of these SCFTs in terms of three-punctured spheres and cylinders, which we do explicitly for A3, and provide tables of properties of twisted defects up through A9. We find atypical degenerations of Riemann surfaces that do not lead to weakly-coupled gauge groups, but to a gauge coupling pinned at a point in the interior of moduli space. As applications, we study: i) 6D representations of 4D superconformal quivers in the shape of an affine/non-affine Dn Dynkin diagram, ii) S-duality of SU(4) and Sp(2) gauge theories with various combinations of fundamental and antisymmetric matter, and iii) realizations of all rank-one SCFTs predicted by Argyres and Wittig.


Introduction and summary
Considerable progress has been made recently in the program of understanding the 4D theories that arise from the compactification of 6D N = (2, 0) theories on a Riemann surface, C, possibly in the presence of codimension-two defects of the (2,0) theories, which correspond to punctures on C [1][2][3][4][5]. 1 Much of the richness of the construction stems from the variety of available defects. When an N = (2, 0) theory of type J = A, D, or E has a nontrivial outer-automorphism group, there exists, in addition to untwisted defects, a sector of twisted defects equipped with an action of a element of the outer-automorphism group of J as one goes around the defect. The general local properties of twisted and untwisted defects were studied in [6]. In particular, the A 2N −1 series has a sector of defects twisted by the Z 2 outer automorphism of A 2N −1 . In this paper, we study the global properties of theories of type A 2N −1 in the presence of such Z 2 -twisted defects. Just as untwisted defects of the A 2N −1 series are classified by embeddings of sl (2) in sl(2N ), twisted defects in this series are classified by embeddings of sl (2) in so(2N + 1). Equivalently, untwisted defects are classified by partitions of 2N , while twisted defects are classified by certain partitions of 2N + 1, called B-partitions. So, for instance, the twisted sector contains a "maximal" twisted puncture, denoted by the B-partition [1 2N +1 ] and with flavour group SO(2N + 1), and a "minimal" twisted puncture, denoted [2N + 1] and with trivial flavour group. The local properties of these and other twisted punctures can be computed following [6]. In this paper we will provide some new, explicit algorithms to make these calculations easier.
One especially interesting twisted defect is the one with B-partition [2N − 1, 1 2 ], which arises from the collision of a minimal untwisted and a minimal twisted defect. Such defect is unique in that it can be continuously deformed into a pair of defects [2N + 1] and [2N − 1, 1]. In the global picture, this property leads to a number of elements that were absent in the untwisted story: • three-punctured spheres that correspond to gauge theories fixed at a point in the interior (not a cusp) of their moduli space.
• cylinders whose pinching leave a gauge coupling at a point in the interior of moduli space.
• cylinders whose pinching decouples a semisimple gauge group.
These phenomena part ways with the usual understanding that a degeneration of a Riemann surface corresponds to the weakly-coupled limit of a simple gauge group in the 4D theory and vice versa. We then study three applications of our constructions: • It is well known that Lagrangian 4D N = 2 superconformal quivers whose gauge group is a product of special-unitary groups can be constructed only in the shapes of (affine and non-affine) Dynkin diagrams of type A-D-E [7]. The A n -shaped quivers were used originally by Gaiotto to deduce local properties of untwisted defects in the A N −1 -series, and are realized as compactifications on untwisted spheres. In this paper we find a realization of the affine and non-affine D n -shaped quivers as compactifications of the A 2N −1 series on twisted spheres. An equivalent expression for the Seiberg-Witten curve for the affine D n -shaped quivers was found long ago by Kapustin [8] from a IIA brane construction. (Quite recently, a uniform way to derive the Seiberg-Witten solutions for these ADE quiver theories were found by [9,10] using instanton calculus.) • We present a full tinkertoy representation of the twisted A 3 theory, and, as an application, study the S-dual frames of SU (4) and Sp (2) gauge theories with matter in the fundamental and antisymmetric representations.
• We show how to construct the rank-one 4D SCFTs studied by Argyres and Wittig in [11,12]. 2 These are SCFTs whose only Coulomb branch operator has scaling dimension ∆ = 3, 4, 6 respectively, and which are not the more familiar Minahan-Nemeschansky theories with E 6,7,8 flavour symmetry. The ∆ = 3 theory will be found in the context of the twisted A 2 theory, although we leave a systematic analysis of the twisted A 2N series for later, due to the subtle issues pointed out in [13]. We will be able to pin down numerical invariants of this ∆ = 3 theory left undetermined in [11,12].
The rest of the paper is organized into two parts. The first part consists of section 2, 3 and 4. In section 2, after recalling the general method for obtaining a 4D theory from a 6D N = (2, 0) theory on a Riemann surface, we describe the algorithms to compute the local properties of twisted punctures of type A 2N −1 , elaborating on [6]. In section 3, we develop the method to identify the behaviour of the theory when two defects are brought together. In section 4, we study atypical degenerations in detail, where the degeneration of the Riemann surface does not correspond to the emergence of a weakly-coupled gauge group.
The second part of the paper deals with applications. In section 5 we show how a D n -shaped quiver gauge theory can be realized in terms of the 6D N = (2, 0) theory of type A 2N −1 on a sphere with twisted punctures. In section 6 we study the S-duality properties of all superconformal SU (4) and Sp(2) gauge theories. In section 7 we discuss rank-one SCFTs and their realizations in terms of 6D N = (2, 0) theory. In appendix A we list all twisted fixtures and cylinders for A 3 , and tabulate the properties of twisted punctures for A 5,7,9 .

4D theories and punctures
In section 2.1 we recall the construction of 4D theories from the compactification of the 6D N = (2, 0) theory of type A 2N −1 on Riemann surfaces with punctures. Section 2.2 through 2.4 detail algorithms to compute local properties of punctures. We show extensive tables of local properties in appendix A. After going over section 2.1, a busy reader can skip the rest of this section, and continue directly to section 3.

Punctures, the fields φ k (z) and the Hitchin field Φ(z)
Consider the 6D (2,0) theory of type A 2N −1 , compactified on a Riemann surface C with a partial twist to preserve supersymmetry [1,14]. We allow for the possibility of having codimension-two defects of the (2,0) theory, localized at punctures on C. This construction leads at low energies to a 4D N = 2 SCFT. We are interested in classifying and characterizing the 4D SCFTs that arise for various choices of C and defects on it.
Usually, the moduli space of the 4D SCFT can be identified with the complex-structure moduli space of C, so that cusps in the latter correspond to weakly-coupled limits of the theory, where a certain gauge group almost decouples. We will see in section 4 that there exist counterexamples to this statement when twisted punctures are included.
The Seiberg-Witten curve Σ of the theory can be realized as a ramified cover of C. To describe Σ explicitly, we should consider the VEVs of certain protected operators in the 6D theory, which, upon compactification on C, give rise to meromorphic k-differentials φ k on C, where the k are the dimensions of the Casimirs of A 2N −1 , i.e., k = 2, 3, . . . , N . The φ k have poles at the positions of the punctures on C. We then have the following equation for Σ: Here λ is the Seiberg-Witten differential, which is a meromorphic 1-form on Σ.
In [6], we discussed a classification of codimension-two defects and how to compute their properties. Defects are grouped into sectors that are in 1-to-1 correspondence with the conjugacy classes of the outer-automorphism group of the simply-laced Lie algebra of the same type as the (2,0) theory that one chooses. In our case, this Lie algebra is A 2N −1 = sl(2N ), which has a Z 2 outer-automorphism group generated by an element o, whose action on the k-differentials is: Then, the sector of untwisted punctures is the one corresponding to the identity element, while the twisted sector corresponds to o. As one goes around a twisted puncture on C, (2.2) tells us that the k-differentials of odd degree k must change sign. Now, untwisted punctures are classified by sl(2) embeddings in sl(2N ), whereas twisted punctures are classified by sl(2) embeddings in so(2N + 1). More practically, recall that sl(2)-embeddings in sl(2N ) are in bijection with partitions of 2N . Similarly, sl(2)embeddings in so(2N + 1) are in bijection with B-partitions of 2N + 1, which are defined as partitions of 2N + 1 where every even part has even multiplicity. For example, [4 2 , 3 3 , 2 6 ] is a B-partition, but [6, 5, 4 2 ] is not.
If z is a local coordinate on C such that the puncture is at z = 0, the k-differentials near z = 0 have the behaviour: We call the set {p k }, for k = 2, . . . , 2N , the pole structure of the puncture. For an untwisted puncture, all the p k should be integer, while for a twisted one, the p k for odd (even) k must be half-integer (integer) because of (2.2). Let us now relate the discussion to the Hitchin system. Following [14], the classical integrable system associated to our 4D N = 2 theories is a Hitchin system on C with gauge group sl(2N ). Let Φ be the Higgs field for the Hitchin system, i.e., Φ is an sl(2N )-valued meromorphic 1-form on C in the adjoint representation of sl(2N ). Then, the Seiberg-Witten curve Σ of this system, (2.1), is given by the spectral curve for the Hitchin system, Thus, comparing with (2.1), we see that the φ k are polynomials in the trace invariants Tr(Φ k ) of the Higgs field. In terms of the Hitchin system, an untwisted defect on C corresponds to a local boundary condition for the Higgs field. Specifically, in local coordinates z on C, let the untwisted puncture be at z = 0. Then, we have where X is an element in sl(2N ) specifying the puncture, and the ellipsis denotes a generic element of sl(2N ). Since Φ is not gauge invariant, the defect is actually characterized by the conjugacy class of X, known as a (co)adjoint orbit in sl(2N ). When the mass parameters of the puncture are set to zero, X is nilpotent, and the orbit is called a nilpotent orbit. Nilpotent orbits in sl(2N ) are classified by sl(2) embeddings in sl(2N ), or, equivalently, by partitions of 2N . If a puncture is labeled by a partition ρ, the nilpotent orbit that defines its boundary condition (2.5) is the one corresponding to the transpose partition ρ T of 2N .
The analogous boundary condition for a twisted puncture was given in [6]. First, decompose the sl(2N ) Lie algebra as a direct sum of eigenspaces of the Z 2 outer automorphism, j = j 1 + j −1 , where j 1 ≃ sp(N ) is the invariant subalgebra. Then, if the twisted defect is at z = 0, the local boundary condition for the Higgs field is Here, X is an element of a nilpotent orbit in sp(N ), and A and A ′ are generic elements of j −1 and sp(N ), respectively. As before, nilpotent orbits in sp(N ) are classified by sl(2) embeddings in sp(N ), or, equivalently, by C-partitions of 2N , which are defined as partitions of 2N where every odd part has even multiplicity. (For example, [6 2 , 3 4 , 2] is a C-partition, but [5 2 , 3, 1] is not.) Then, a twisted puncture in the A 2N −1 series is labeled by a B-partition ρ of 2N + 1, but its Higgs-field boundary condition is given by a C-partition ρ ′ of 2N . There is a map, called the Sommers-Achar map [15][16][17], which is a generalization of the Spaltenstein map on nilpotent orbits, which gives us the Hitchin-system data associated to ρ: (2.7) Here, C(ρ) is a discrete group. Then, X is the nilpotent element ρ ′ (σ + ), seeing ρ ′ as an sl(2) embedding in Sp(N ). In [6], ρ ′ was called the Hitchin pole of the puncture labeled by the Nahm pole ρ. ρ ′ is given by the C-collapse of the reduction of the transpose of ρ (a 2N + 1 partition) to a 2N partition; see section 2.2 and section 3.4.4 of [6].

Local properties of punctures
The local properties of a twisted puncture that we can compute are: p k , or to the fact that a leading coefficient can be expressed in terms of more basic gauge-invariants, as we will see in section 2.4. (In principle, subleading coefficients may have been constrained too, but it turns out that this does not occur.) Once the local form of the Higgs field Φ for a specific puncture is known, as in (2.5) and (2.6), one can find the local form of the k-differentials from (2.1) and (2.4), read off the the pole structure {p k }, find the constraints, and compute the {n k }.
However, carrying out this "honest" procedure is quite tedious in practice. In what follows, we describe algorithms to compute these properties directly from the B-partition, which we found after looking at a large number of examples. First, in section 2.3, we explain how to calculate the {n k }, and then, in section 2.4, how to compute the constraints. Once these are known, the pole structure {p k } can be easily reconstructed. We will see that the only twisted defect that gives rise to a Coulomb branch operator of dimension two is the one with B-partition [2N − 1, 1 2 ]. This occurs through a constraint of the form c This particular puncture will play an important role in section 4.
For untwisted punctures in the A series, it is well known that there are no constraints at all, and so the pole orders {p k } are exactly the same as the {n k }, for each k. (By contrast, untwisted punctures in the D series generically do exhibit constraints [5].) The Lie algebra of the global symmetry group G flavour of a twisted puncture labelled by the embedding ρ : su(2) → so(2N + 1) is the commutant of ρ(su(2)) in so(2N + 1). It is easier to give a formula for G flavour in terms of the B-partition p corresponding to ρ: where l runs over the parts of the partition p, and n l is the multiplicity of l in p. For even l, n l must be even because p is a B-partition, so the formula above makes sense. (In our notation, Sp(1) ≃ SU(2).) As for the contributions to n h and n v (and thus a and c), these can be easily computed from the embedding ρ : su(2) → so(2N + 1). The formulas were given in [6], Here, ρ A and ρ B are the Weyl vectors of A 2N −1 and B N , respectively; h = ρ(σ 3 ) is the Cartan of the embedded su(2), and we have decomposed g = so(2N + 1) = r∈Z+1/2 g r , where g r is the eigenspace of h with eigenvalue r. The contributions to n h and n v for the twisted sectors of the A 3,5,7,9 theories are given in appendix A.

Graded Coulomb branch dimensions
Consider a twisted puncture in the A 2N −1 theory, specified by a B-partition p of 2N + 1. We want to compute the contributions {n k } to the dimensions of the graded Coulomb branch. The formula for the {n k } is most easily expressed in terms of a number of auxiliary sequences, which we now define. Let q = p t be the transpose partition to p. First, let us define a sequence ν by Next, let s be the sequence of partial sums of q, Finally, define a sequence r of "corrections" by r k = 1 if k ≤ N and 2k / ∈ s, 0 otherwise. (2.12) Then, the contribution n k for the twisted puncture with B-partition p is (2.14) Similarly, the {n k } for the full (or maximal) twisted puncture, p = [1 2N +1 ], are

General structure of constraints
The structure of constraints for twisted punctures in the A series is relatively simple. These constraints satisfy some properties: • They are polynomials in the leading coefficients c  • The polynomials f ii) c iv) c 3 c 2 .
The first and third examples are "squares", while the second and fourth are "cross-terms". Also, the first and second examples involve only the c (k) l , while the third and fourth involve also new parameters a (k) l . We call constraints that do not introduce any new parameters, such as the first two examples above, c-constraints. A c-constraint of scaling dimension k (which necessarily has pole order l = p k ) tells us that the leading coefficient c (k) l is dependent on others, and so the local contribution to n k should be reduced by one.
By contrast, the third example, of the form c is the square of another, more basic gauge-invariant parameter, a (k) l . Thus, it effectively trades a parameter of scaling dimension 2k by a parameter of scaling dimension k; in other words, the contribution to n 2k is reduced by one, while the contribution to n k is raised by one. We call this type of constraint, which introduces a new parameter, an a-constraint.
Finally, the fourth example is a cross-term involving the parameter a (k) l . However, the a (k) l will already have been introduced by an a-constraint as in the previous paragraph. Hence, this cross-term constraint should be taken to be a c-constraint, not an a-constraint. Generically, for every a (k) l , there will be exactly one a-constraint (a square in a

Number of constraints
Now, for a given twisted puncture, let us explain the rule to find at which scaling dimensions k there exists a constraint. Denote by p the B-partition of 2N + 1 that labels our twisted puncture. Consider, as before, the transpose partition, q = p t , and let s be the sequence of partial sums of q, as in (2.11) above. We will see that s contains all the information about constraints.
Let us first note that a B-partition always has an odd number of parts, so suppose our B-partition p has 2l + 1 parts, and let p 2l+1 be the last part of p. Then, an a-constraint of scaling dimension k exists if and only if: 1. k belongs to s.

k is even.
3. k is not a multiple of 2l + 1.
If k = 2m satisfies these conditions, the local contribution to n 2m should be reduced by one, and the contribution to n m should be raised by one.
Similarly, a c-constraint of scaling dimension k exists if and only if: 1. k belongs to s.
2. If k is odd, it must satisfy a "cross-term" condition. Let j be the unique index such that k = s j . Then: 1) s j must not be the last element of s; 2) both s j−1 and s j+1 must be even, s j−1 = 2u and s j+1 = 2v; and 3) s j = u + v.
3. If k is even, it must be a multiple of 2l + 1.
4. If k is a multiple of 2l + 1, that is, k = r(2l + 1) with r integer, then If k satisfies these conditions, the local contribution to n k should be reduced by one.
61/2 , with the same dimension and pole order as the leading coefficient c (33) 61/2 , these are independent.

Explicit form of constraints
Now, the rules described above (to compute the dimensions at which a-and c-constraints occur) are sufficient for most purposes, but if we want to know what the constraints look like more specifically, which we need to compute explicit Seiberg-Witten curves, we should study the constraint structure of twisted punctures a little more systematically. We will do so below.
Recall our B-partition p has 2l + 1 parts, p = {p 1 , . . . , p 2l+1 }. Then, q must be of the form Hence, the first p 2l+1 parts of the set of partial sums, s, are multiples of (2l+1) in arithmetic progression: s = [(2l + 1), 2(2l + 1), 3(2l + 1), . . . , p 2l+1 (2l + 1), . . . ] (2.17) (By construction, the entry to the right of p 2l+1 (2l + 1) cannot be a multiple of 2l + 1.) This block of multiples of 2l + 1 in s will be important, since it gives rise to a particular set of c-constraints for p. So, let us look at it in detail. Consider the first 2 p 2l+1 2 multiples of 2l + 1 in s, split into two groups: For completeness, let us call s ′′′ the set of entries of s that are not in s ′ or s ′′ , so s = s ′ ∪ s ′′ ∪ s ′′′ is a disjoint union. Notice that if p 2l+1 is odd, the term p 2l+1 (2l + 1) is in s ′′′ . This term never gives rise to a constraint.
Entries in s ′ . None of the entries in s ′ correspond to constraints. Rather, they can be used to define certain quantities that make c-constraints more transparent; see the example of the minimal puncture, p = [2N + 1], below.
Entries in s ′′ . Each entry in s ′′ can be interpreted as a dimension for a c-constraint. Let us look at these constraints in more detail. We study first the even entries. Let 2k be in s ′′ . The corresponding c-constraint is, schematically, a square: l/2 (c) is a polynomial in leading coefficients, of total dimension k and total pole order l = p 2k (e.g., f ). The ellipsis above (and in the rest of this subsection) stands for possible cross-terms, which are of the form (2.20) Such a term would arise if and only if there exist c-constraints of dimensions 2k ′ and 2k ′′ , and if k ′ + k ′′ = 2k and l ′ + l ′′ = 2l. On the other hand, the odd entries, say 2k + 1, of s ′′ always yield c-constraints that are sums of cross-terms, as (2.20), but with k ′ + k ′′ = 2k + 1.
Entries in s ′′′ . Let us now study the constraints in s ′′′ . Again, let us look at even and odd entries separately. Each even entry, 2k, in s ′′′ is the dimension of an a-constraint, Finally, let us look at the odd entries, 2k + 1, in s ′′′ . If 2k + 1 satisfies the requirements of section 2.4.2, it yields a cross-term c-constraint involving parameters introduced by a-constraints, c where u + v = 2k + 1 and l = l ′ + l ′′ . Also, if the first c-constraint dimension in s ′′′ is odd, the c-constraint will include a "mixed" cross-term of the form To write the c-constraints specifically, we define auxiliary quantities r k , for 0 ≤ k ≤ N , by r 0 ≡ 1, r 1 ≡ 0, and the rest by p k is the sum of all terms of the form (r j ) 2 or 2r j r j ′ , with j, j ′ < k, of total scaling dimension k and total pole order p k . Then, expressing the c (k) p k back in terms of the r j , for 0 ≤ j ≤ k ≤ N , reveals a nice pattern of squares of cross-terms that should be completed, in a unique way, by the sought c-constraints. For instance, for N = 5, we define .

(2.24)
Then, we can write: 5/2 = 2r 5 r 0 +2r 3 r 2 +2r 1 r 4 , c 7/2 = 2r 3 r 4 + 2r 2 r 5 , c Here, the expressions of scaling dimension k for 0 ≤ k ≤ 5 are equivalent to (2.24), while those with 6 ≤ k ≤ 10 are the actual c-constraints. Thus, introducing the r k makes clear what the c-constraints should be. Now, let us discuss the puncture p = [2N − 1, 1 2 ]. We find that there are a-constraints (c-constraints) for every even (odd) scaling dimension k in the range 4, 5, . . . , 2N . These constraints follow a pattern of squares and cross-terms in a puncture is the only one with an a-constraint of scaling dimension four, that is, the only one that gives rise to an independent parameter a (2) of scaling dimension two.

Collisions of punctures
In this section we study what happens when two or more punctures collide. We call this process the operator product expansion (OPE) of punctures. In section 3.1 and section 3.2, we discuss the overall strategy for analyzing the OPE, by first considering the OPE on an infinite plane, and then on a compact curve. We then describe an explicit algorithm to compute the OPE in section 3.3.

OPE of punctures on a plane
So far we have studied how to compute the properties of a single puncture. Let us now see what happens if two or more punctures come close together. First, we would like to study the simpler case of a non-compact Riemann surface, the complex plane. Consider a sixdimensional space of the form R 4 × C. We denote by z the coordinate on C, and consider k punctures of types p 1 , p 2 , . . . , p k to be localized, respectively, at z = z 1 , z 2 , . . . , z k . Now, at very large |z|, the system looks as if consisting of: • a puncture q at z = 0, with flavour symmetry F , • a 4D N = 2 superconformal theory X, which depends on the types and positions of the k punctures, and such that a certain subgroup H of its global symmetry group, G X , can be identified with a subgroup of F , and • a dynamical gauge multiplet for H, with coupling constant τ depending on the types and positions of the k punctures, which couples X to q.
We call this process the operator product expansion (OPE) of the k punctures, and we call X the coefficient of the OPE. We schematically represent the outcome of the OPE as If q is the full puncture and H = G, the theory X is the same as the 4D theory obtained by compactifying the 6D theory on a sphere, with punctures of type p i at z = z i , and a full puncture at z = ∞. Otherwise, we say that the theory X is the 4D theory "obtained by compactifying the 6D theory on a sphere, S, with punctures of type p i at z = z i , and an irregular puncture at z = ∞, determined by the choice of p i ," and say that JHEP05(2015)075 "the gauge group H arises from the cylinder connecting the irregular puncture with the regular puncture of type q." 3 We denote such irregular puncture by the pair (q, H), and, if there are inequivalent embeddings H ֒→ F , we add a label to distinguish which embedding we mean; see section 3.2.3 for an example. We call q the "regular puncture conjugate to" the irregular puncture (q, H). While the detailed properties of the theory, X, depend on the punctures, p i (and the various cross-ratios of their positions), certain features are encoded purely in the pair (q, H). For instance, H, seen as a subgroup of the global symmetry group G X of the theory X, has some level k ≥ 0. (k = 0 if and only if X is the empty theory.) This level is strictly determined [5] by demanding that the H gauge theory on the cylinder Similarly, the local contribution of the irregular puncture to n h , n v and to the graded Coulomb branch dimensions of X are determined by the pair (q, H) [5].

Degeneration of a curve via the OPE
Let us now consider the OPE on a compact curve. Let C be a sphere, with k + k ′ regular punctures of types p 1 , . . . , p k ; p ′ 1 , . . . , p ′ k ′ . We assume that the punctures are such that all the graded Coulomb branch dimensions are non-negative. (Otherwise, the theory is "bad", and taking the 4D limit is a more delicate issue.) Now consider the limit where C degenerates into two spheres, We would like to understand the behaviour of the 4D theory in this limit. We proceed as follows: • Replace the punctures p 1 , . . . , p k with their OPE, as in section 3.1, obtaining a regular puncture q, a gauged subgroup H of the flavour symmetry of q, and the 4D theory X, which is the 4D limit of a sphere with p 1 , . . . , p k plus an (ir)regular puncture (q, H).
• Then we have the following system consisting of where I(q, q ′ ) is a sphere with two regular punctures of type q and q ′ , respectively.
• As explained in [18], a sphere with two regular punctures is a supersymmetric hy-perKähler non-linear sigma model with global symmetry F × F ′ , where, in our case, we gauge the subgroup H × H ′ ⊂ F × F ′ . Any point on the target space of the non-linear sigma model breaks F × F ′ to the stabilizer subgroup, F ′′ and hence the gauge symmetry H × H ′ is always Higgsed to H ′′ ⊂ F ′′ .
• Sometimes, the D-term and the F-term constraints for H ′′ force the theory X coupled to q via H to be Higgsed to a theory Y . Similarly, the theory X ′ coupled to q ′ may be Higgsed to a theory Y ′ .
• In the end, we have a 4D system of the form: Now, a sphere with k regular punctures and an irregular puncture has a degeneration where we consecutively collide two punctures, so that the resulting 4D theory consists of several three-punctured spheres coupled to each other. These three-punctured spheres, which we call fixtures [4,5], contain either • three regular punctures, or • two regular punctures and one irregular puncture.
A table of all possible fixtures makes finding the 4D description of an arbitrary degeneration a simple task. Let us illustrate these ideas with a few examples, all with untwisted punctures for simplicity.

Example 1
Consider the A 2N −1 theory compactified on a 4-punctured sphere, with punctures • The OPE of p 1 with p 2 is a full puncture, [1 2N ], coupled via H = Sp(N ) to the theory, X, which is 4N free hypermultiplets transforming as 2 copies of the fundamental 2Ndimensional representation of Sp(N ).
Here the symbol stands for the 2(N − 1)-dimensional fundamental representation of
• The OPE of p 3 and p 4 yields the full puncture, q ′ = [1 5 ], coupled to the theory Here R 2,5 is a non-Lagrangian SCFT discussed in [4]. Note that, since q ′ is the full puncture and H ′ is G, the 3-punctured sphere corresponding to X ′ contains three regular punctures, • In between, we have the 2-punctured sphere with one full puncture and one [2, 1 3 ] puncture. This Higgses H × H ′ = SU(2) × SU (5) down to H ′′ = SU (2). However, in order to satisfy the F-term equations, the theory X ′ is Higgsed down to the theory The end result is , SU(2) SU (2) empty (E 6 ) 6 + 1(2) + 5 (1) an SU(2) gauging of the (E 6 ) 6 SCFT, with an additional doublet and 5 free hypermultiplets. Note that, in this case, the cylinder connects an irregular puncture with its conjugate regular puncture.

Example 3
Now, let us turn to an example from the D 4 theory. Consider the 4-punctured sphere Here, each "very even partition" (e.g., [2 4 ]) corresponds to two nilpotent orbits in so (8), and our sphere includes one of each type (indicated by the red/blue colour); see [5]. When we take the OPE of p 1 with p 2 , we obtain the full puncture, q = , coupled to X = the (E 7 ) 8 SCFT, via H = Spin(7) (and similarly for the OPE of p 3 with p 4 ). However, there are three inequivalent embeddings of Spin(7) ֒→ Spin (8), depending on which of the three 8-dimensional irreducible representation of Spin(8) decomposes as 7 + 1. We can indicate this choice by putting a subscript on H, or (in the notation of [5]) by colouring the Young diagram corresponding to q: , Spin (7)) = ( , Spin(7) 8c ) In the notation of (3.2), we have H = Spin(7) 8s and H ′ = Spin(7) 8c , and the two-punctured sphere, I( , ) Higgses H × H ′ = Spin (7) 8s × Spin (7) 8c down to the common subgroup, G 2 . So the final 4D description of this limit is , Spin (7) , where, as in section 3.2.1, we have a cylinder connecting two irregular punctures.

Determining the OPE via the Higgs field
In light of section 3.1 and 3.2, we would like to study the basic problem of two punctures p 1 and p 2 colliding on a plane. We have seen that in the collision limit, an irregular puncture (q, H) arises, which is connected to a regular puncture q by a cylinder with gauge group H. Let us discuss how to find q and H.
To determine q, we construct a solution to the Higgs field on the plane that includes p 1 and p 2 , and compute the residue that arises in the collision limit. This residue provides the Higgs-field boundary condition for q. Thus, one can determine the Nahm pole for q, e.g., by looking at the degeneracy of the mass deformations in the residue. Also, the number of independent mass deformations is equal to the rank of H.
To gather more information about H, we consider the k-differentials φ k , and take the limit where the punctures collide, which reveals the scaling dimensions of the Casimirs of H. Knowing these usually suffices to identify the gauge group. Only in a handful of cases, often to distinguish Sp(k) from SO(2k + 1), must one do further consistency checks, such as computing the matter representation for the fixture that arises in the degeneration limit, and corroborating that it provides the right contribution to the beta function of H.
Because of these observations, in the next subsections we will study the Higgs field on a plane with two punctures, in the limit where these collide. Later, in section 3.3.4, we will do the same for k-differentials. But before doing this, let us briefly discuss a situation that will arise often.
Consider C to be the complex plane or a sphere, with complex coordinate z, and put k punctures, p 1 , . . . , p k , on C. Let the positions of the punctures be λz 1 , . . . , λz m , z m+1 , . . . z k , so that we can collide the first m of them by taking the limit λ → 0. Now consider a meromorphic k-differential on C of the form where α, s, r 1 , . . . , r k are rational numbers; A is a coefficient. This is a typical term in a k-differential, including the case of the Higgs field (k = 1). In the λ → 0 limit, we get C in the presence of the k − m punctures p m+1 , . . . , p k , plus a new puncture, q, at z = 0.
The λ → 0 limit may also be represented by the conformally equivalent picture of a sphere C ′ that bubbles off C, containing the m punctures p 1 , . . . , p m , plus the irregular puncture (q, H). Such picture is obtained through the change of variables z = λ/z ′ . Then, requiring that (3.4) have a finite limit as λ → 0 in the z ′ coordinates puts a lower bound on α, (3.5) We will use this result quite often. If the bound is not saturated, the k-differential simply vanishes, in the λ → 0 limit, on both C and C ′ . On the other hand, if the bound is saturated, we have three possibilities when λ → 0: In most cases (such as in the following subsections), the coefficient A in (3.4) will represent a physical degree of freedom, and so it should not be lost in the λ → 0 limit; therefore, it will be desirable that the bound be saturated. Case 1 (2) corresponds to A being a degree of freedom for the theory on C ′ (C), whereas case 3 corresponds to A being a degree of freedom of the gauge group on the cylinder, which in the λ → 0 limit looks like a mass deformation on both C and C ′ . However, in a few cases where the coefficient A carries redundant information (because of local constraints), consistency will require that the bound not be saturated. We will see such cases in section 4.

Untwisted-untwisted
Consider now the Higgs field Φ on a plane with two untwisted punctures of type p 1 and p 2 at positions z = 0 and z = λ, respectively, where z is the coordinate on the plane. Let A and B be representatives of the (massless or mass-deformed) adjoint orbits in sl(2N ) corresponding to p 1 and p 2 , respectively. Then, we can write an ansatz, where P (z) is a power series in z whose coefficients are generic elements of sl(2N ). P (z) simply represents the infinite degrees of freedom contained in the plane. At finite λ, the expansions of Φ(z) near p 1 and p 2 are, respectively, In the limit λ → 0, a new untwisted puncture, q, arises at z = 0. The expansion of Φ(z) near this point is This is the expected expansion for an untwisted defect. Thus, the Higgs field boundary condition for the new puncture q is given by A + B. Notice that (3.6) saturates the bound of (3.5). In the λ → 0 limit, we have the complex plane in the presence of just the new puncture, q. The conformally-equivalent picture is that of a fixture that bubbles off the plane, containing the two original punctures, p 1 and p 2 , plus the irregular puncture, (q, H). The Higgs field for the fixture is where we used z = λ/z ′ in (3.6) and took the λ → 0 limit. The punctures q, p 1 , p 2 are at z ′ = 0, ∞, 1, respectively. Notice that the Riemann-Roch theorem for a 3-punctured sphere requires only two coefficients, not three, for Φ(z ′ ). In other words, in a fixture, the choice of representatives, A and B, for the adjoint orbits for two punctures, p 1 and p 2 , completely determines the adjoint orbit (that is, its mass deformations), plus a representative of such orbit, for the third puncture, q.

Twisted-twisted
Let us now take two twisted punctures of type p 1 and p 2 at positions z = 0 and z = λ on a complex plane with coordinate z. Let A and B be representatives of the (massless or massdeformed) adjoint orbits in Sp(N ) corresponding to p 1 and p 2 , respectively. Recall the decomposition of sl(2N ) in eigenspaces of the Z 2 -outer automorphism, sl(2N ) ≃ sp(N ) ⊕ o −1 . Then, we can write the following ansatz for the Higgs field, where D is a generic element of o −1 , and P (z) and Q(z) are power series in z whose coefficients are, respectively, generic elements of sp(N ) and o −1 . At finite λ, the expansions of Φ(z) near p 1 and p 2 are, respectively, These expansions are of the expected form for twisted defects. Now, in the collision limit λ → 0, a new untwisted puncture, q, arises at z = 0. The expansion of Φ near q is This is, again, the expected expansion for an untwisted defect. Thus, q has Higgs field residue A + B + D, with D generic in o −1 .
For completeness, we show the Higgs field for the fixture in the conformally equivalent picture, as before, (3.14)

Twisted-untwisted
Finally, consider a twisted and an untwisted puncture, of types p 1 and p 2 , respectively, at positions z = 0 and z = λ. Let A and B be representatives of the (massless or massdeformed) adjoint orbits corresponding to p 1 and p 2 , respectively. Notice that A is in sp(N ), while B is in sl(2N ). Let us decompose B as B = B 1 + B −1 , where B 1 is in sp(N ) and B −1 is in o −1 . Then, we can write an ansatz for the Higgs field, where P (z) and Q(z) are power series in z whose coefficients are, respectively, generic elements of sp(N ) and o −1 . The expansions of Φ(z) near p 1 and p 2 are, respectively, (3.16) Again, these expansions have the expected forms. Now, in the limit λ → 0, a new twisted puncture, q, arises at z = 0. The expansion of Φ(z) near q is This expansion has the correct form for a twisted defect. Thus, the Higgs field boundary condition for q is given by A + B 1 , with B 1 the projection of B to sp(N ). Finally, the Higgs field for the fixture in the conformally equivalent picture is (3.18)

Degenerating k-differentials
Let us discuss how to find the scaling dimensions of the VEVs for the gauge group H that arises when two punctures p 1 and p 2 on a plane collide. In most cases, these provide enough information to determine H. The natural way to find such VEVs is to use (2.4) to compute the k-differentials from the Higgs field residue of the new puncture, q. If q is at z = 0, we have, in principle, a gauge group VEV u k of scaling dimension k if However, some of these u k may not be independent. If p 1 and p 2 have mass deformations, for instance, the u k might contain combinations of these masses. If a parameter u k vanishes when we turn off the the masses, then such u k is not an actual gauge group VEVs. Also, the u k might depend on each other. Hence, it is convenient, for the purpose of finding the gauge group, to consider the OPE with massless p 1 and p 2 . It might also be useful to study the k-differentials before the collision, taking into account whichever c-constraints and a-constraints the two punctures have, and then take the degeneration limit. Let us also briefly discuss the basic problem of the degeneration of a sphere with n 1 +n 2 massless punctures, in which we collide the first n 1 punctures. Each k-differential contains 1 − 2k + n 1 +n 2 i=1 p (i) k terms of the form (3.4). We want to find conditions for a gauge group VEV term (3.19) to exist at every k. Note that if k is odd, and we are colliding an odd number of twisted punctures, there cannot be a gauge group term because the power s in (3.19) must be an integer. So, if we collide an odd number of twisted punctures, 4 the discussion below is restricted to even k.
If none of the punctures satisfy constraints, then, at every k, there will exist a gauge group term (3.19) if and only if t k ≡ −k + n 1 i=1 p (i) k ≥ 0 and t ′ k ≡ −k + n 1 +n 2 j=n 1 +1 p (j) k ≥ 0. If there are c-constraints and a-constraints, these will act on the parameters of the kdifferential on the side of the degeneration where they exist, and so may constrain the gauge group VEV. However, if t k ≥ n c + n a , where n c and n a are, respectively, the number of c-constraints and a-constraints of dimension k for the punctures on one side of the degeneration, and if a similar condition holds for the other side of the degeneration, then the gauge group term will not be affected by the constraints. But if these conditions do not hold, one needs to analyze the k-differential in detail to see how the gauge group VEVs are affected. Now we wish to illustrate our techniques with some examples. To write residues explicitly, we need to use a basis. We use the embedding of Sp(N ) in sl(2N ) of [19], where A, B and C are n × n complex matrices, and B and C are furthermore symmetric. We also take from [19] the nilpotent orbit representatives of sl(2N ) and Sp(N ).

Example 1
Let us consider again the untwisted example of section 3.

Example 2
Let us study the case of a five minimal twisted punctures, [2N + 1], on a plane, and study the consecutive collisions of two punctures. We do this to illustrate the rules for combining residues.
Colliding two minimal twisted punctures yields an untwisted puncture with residue equal to a generic element in o −1 . This can be diagonalized to an element of the form diag(m 1 , m 2 , . . . , m N ; m 1 , m 2 , . . . , m N ), (3.23) with N i=1 m i = 0. Since the 2N eigenvalues are split into pairs of identical elements, this residue should be a mass-deformed representative of the untwisted [2 N ] puncture. This puncture has SU(N ) global symmetry group. The fact that there are N masses m i adding up to zero suggests that the whole SU(N ) group is gauged. This can be checked by computing the k-differentials. In fact, to leading order we have, for k = 2, 3, . . . , 2N , where the two colliding [2N + 1] punctures are at z = 0, λ. All of the u k survive the λ → 0 limit, and so each should become a Casimir of the gauge group. However, only the u k for k ≤ N are independent because of the constraints from the minimal twisted punctures. (Actually, since there are two [2N + 1] punctures, the subleading coefficients in the expansion above for k > N are also constrained.) Thus, the gauge group has only N − 1 Casimirs of scaling dimensions 2, 3, . . . , N . Hence, it must be SU(N ). Now let us find the OPE of the [2 N ] puncture with the third [2N + 1] puncture. Our prescription tells us that the residue of the new puncture will be the Sp-projection of the residue of the [2 N ] puncture. 5 The diagonalized form of the new residue is diag(r 1 , r 2 , . . . , r N ; −r 1 , −r 2 , . . . , −r N ), (3.25) with N i=1 r i = 0. For N = 2, this residue takes the more particular form diag(r, r, −r, −r). So, for N = 2, this is the [2 2 , 1] puncture, and for N ≥ 3, it is the [1 2N +1 ] puncture.
To find the gauge group, we can either use three colliding [2N + 1] punctures, or the [2 N ] puncture colliding with a [2N + 1] puncture. Let us use the first. A [2N + 1] puncture has pole structure p k = k/2, and satisfies a c-constraint for k = N + 1, N + 2, . . . , 2N . Plus, for k = 2, . . . , N , we have t k = 3 i=1 p i k − k = k/2 ≥ 0, so we have an unconstrained VEV for each even k in this range. On the other hand, at each k = N + 1, . . . , 2N , we have three c-constraints, so we have t k − n c = 3 i=1 p i k − k − 3 = k/2 − 3, and we can be sure we have an unconstrained VEV for every even k in this range such that k ≥ 6. So, for N ≥ 5, the gauge group is just Sp(N ), while for N = 2, 3, 4, we need to check by hand. For N = 2, we have k = 4 < 6, so we only have one VEV of dimension 2, and the gauge group should be SU (2). For N = 3, we have k = 4 < 6, but k = 6 ≥ 6, so we have VEVs of dimensions 2 and 6, and the gauge group should be G 2 . For N = 4, we have k = 6 ≥ 6 and k = 8 ≥ 6, so we have an Sp (4)

Atypical degenerations
The conventional understanding of Gaiotto duality is that one starts with a Riemann surface, C, with punctures, and, in any degeneration limit of C, a weakly-coupled gauge group, G, in the low-energy 4D theory arises. There is a specific connection between the plumbing fixture q of the degenerating cylinder and the weak gauge coupling τ for G, q = e 2πiτ (4.1)

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So, the pinch limit q → 0 of the surface corresponds to the weakly-coupled limit τ → i∞ for the gauge group. Besides, the gauge-invariant VEVs constructed from the scalars in the G-vector multiplets can be similarly assigned to the cylinder: upon degenerating the curve, the VEVs of G become mass-deformations of the new punctures that appear on both sides of the degeneration. Finally, upon complete degeneration, the G-vector multiplets completely decouple; they are not present on either side of the degeneration.
In this section we want to study certain degenerations involving curves with certain combinations of twisted and untwisted punctures A 2N −1 series where the picture explained above relating cylinders and gauge groups does not hold. 6 Our goal is to understand to what extent the conventional picture of the previous paragraph is still correct, and how it should be modified when our curve contains these "dangerous" combinations of punctures. Happily, "atypical" degenerations are actually rare.
By "atypical" degenerations in the twisted A 2N −1 series we refer to either of the following situations: • A degeneration brings a certain gauge coupling to a point in the interior of moduli space of couplings, instead of a weakly-coupled cusp. This interior point can either be a strongly-coupled point, or a point where two gauge group couplings become equal.
• A degeneration brings not only the corresponding gauge group to its weakly-coupled cusp, but also other gauge groups (adjacent to, but not directly localized at the degenerating cylinder) to weakly-coupled cusps.
Fortunately, these atypical degenerations seem to occur only when a sphere, containing certain combinations of punctures, bubbles off a generic surface. Furthermore, for the bubbling sphere to be "atypical", there is a bound on the number of punctures it may contain, as well as a restriction on the types of punctures. (These can only be minimal twisted or minimal untwisted.) Thus, these "atypical" spheres are easy to classify. Let us study these "atypical" spheres one by one. = a 2 in the [2N − 1, 1 2 ] puncture introduces a Coulomb branch VEV a of scaling dimension two. We want to argue that such a fixture represents a gauge theory whose gauge coupling τ is locked at the Z 2 -symmetric point of its moduli space, τ = i, and that the Z 2 action can be identified with the disconnected part of the flavour symmetry O(2) of [2N − 1, 1 2 ]. A 3-punctured sphere has no moduli, so the fact that a gauge coupling is frozen at a point in coupling-constant moduli space is the only way that we could have a gauge theory on a 3-punctured sphere. Now, it is not obvious that such point in the moduli space should lie in its interior, or that it should take the value τ = i. We will verify below these assertions by studying the Seiberg-Witten curve for the fixture.
Let the punctures [2N − 1, 1 2 ], U , and T be at the positions z = 0, 1, ∞, respectively. Since we are just interested in seeing at which point in gauge-coupling moduli space the theory is, we remove all the unnecessary degrees of freedom, such as mass deformations and Coulomb branch VEVs of scaling dimension different from two. Then, the only surviving k-differential is φ 4 , which includes the square of the Coulomb branch parameter a. Notice that the 2-differential φ 2 vanishes because we have only three massless punctures. So, the Seiberg-Witten curve for the reduced theory is: The factor y 2N −4 tells us that the original 2N -sheeted cover Σ of the fixture splits into 2N − 4 unramified branches plus a four-sheeted cover, Σ ′ . Let us dispose of the unramified branches. The Seiberg-Witten curve factorizes: This expression tells us that Σ ′ globally splits into two double covers which differ only by the choice of sign in a. Let us explore the first factor in (4.3). Consider the transformation z = t 2 , y = y ′ /2t, which preserves the Seiberg-Witten differential, λ = ydz = y ′ dt. Then the first factor takes the form y ′2 − 4a t(t − 1)(t + 1) = 0 (4.4) Now, this is the Seiberg-Witten for a four-punctured sphere in the untwisted A 1 theory, with punctures at t = 0, 1, −1, ∞, which represents the SU(2) N f = 4 gauge theory. If the curve above had really arisen from a four-punctured sphere, we would identify the marginal coupling q of the SU(2) gauge group with the cross-ratio x of the four punctures. But here, since we started with a fixture, we are not allowed to vary the cross-ratio; the curve we obtain is fixed at the cross ratio, x = −1. This value corresponds to the Z 2 -orbifold point, τ = i, in gauge-coupling moduli space. What about the second factor in (4.3)? It clearly represents again the SU(2) gauge theory, but with the other choice of sign for a. The origin of this second factor is the aconstraint of the [2N − 1, 1 2 ] puncture, which does not fix the sign of a. This second copy is not a second, independent, SU(2) gauge theory, since its degrees of freedom simply mirror those of the first factor up to a sign. So, the original fixture we started with represents a single copy of the SU(2) gauge theory at the τ = i point in moduli space. This suggests that the original, good A 2N −1 fixture can be interpreted to contain a gauge group G with gauge coupling τ = i. This interpretation is confirmed by computing the total number of hypers and vectors for this fixture, as well as the representations for the matter, and is consistent with S-duality in all examples we studied. via an "empty" cylinder, to form the following four-punctured sphere: Gauge theory at the Z 2 -symmetric point In this degeneration, the cross-ratio x of this four-punctured sphere controls the perturbation of the gauge coupling τ away from the Z 2 symmetric point, τ = i. Locating the punctures [2N + 1], T , [2N − 1, 1], U at z 1 , z 2 , z 3 , z 4 respectively, let We will find that the relation between the cross-ratio of our original twisted 4-punctured sphere and the SU(2) gauge coupling q is given by At x = 0, the gauge coupling becomes q = −1, i.e., the Z 2 orbifold point, which is consistent with what we found for the gauge-theory fixture alone. Note that before the degeneration, the flavour symmetry comes from the puncture [2N − 1, 1] which has U(1) symmetry. After the degeneration, the flavour symmetry comes from [2N − 1, 1 2 ] which has O(2) symmetry. This contains the original U(1) together with its outer automorphism. The appearance of a square root of x in (4.5) means that the marginal-coupling moduli space of the theory, which we will denote by X 4 , is not the complex structure moduli space of the punctured sphere, M 0,4 , but a double cover of it. X 4 is parametrized by w, rather than the cross-ratio, x. This feature recurs later in the discussions in section 5. The Z 2 deck transformation for this cover, w → −w, implements the Z 2 S-duality transformation q → 1/q. (4.6) on the family of gauge theories, of which the theory at q = −1 is a fixed-point. Moreover, this Z 2 S-duality transformation is accompanied by a Z 2 outer automorphism that acts on the global symmetry group of the theory. In particular, it acts as a Z 2 automorphism of the SCFT at q = −1.

Derivation
Using the prescription of section 3.3.3, one can check that the OPE of the massless punctures of types [2N + 1] and [2N − 1, 1] yields the massless [2N − 1, 1 2 ] puncture. This is a remarkable property. Usually, the OPE of two massless punctures yields a mass-deformed new puncture; these masses give rise to the VEVs for the gauge group on the cylinder. In general, the masses of the new puncture are encoded in the choice of representatives of the orbits for the two original punctures, as emphasized in footnote 5. In our case, the [2N + 1] puncture has only one possible representative, the zero element. However, one still has a choice of representative for the [2N − 1, 1 Let us now derive the relation (4.5) explicitly by studying the Seiberg-Witten curve of the 4-punctured sphere. We put the punctures of types [2N + 1], [2N − 1, 1], U , T , at the positions z = 0, x, 1, ∞, respectively, and consider the x → 0 limit. As before, we discard all parameters irrelevant to the problem, that is, all mass deformations and Coulomb branch VEVs of scaling dimension different from two. So, the only non-zero kdifferentials will be φ 2 and φ 4 . In eliminating Coulomb branch VEVs, we have solved the constraints of the [2N + 1] puncture by imposing the relation c (which is itself a constraint for N = 2) and setting to zero any other parameters. Thus, effectively, we have reduced a problem in the A 2N −1 theory to one in A 3 . The k-differentials are: (4.7) The bound in (3.5) implies that γ ≥ 0 and β ≥ 0, while the constraint requires u 4 = −(u 2 ) 2 and β = 2γ − 1. Hence, the bounds are refined to γ ≥ 1/2 and β ≥ 0. But we cannot have γ > 1/2, β > 0 because then both u 2 and u 4 would disappear from either side of the degeneration when x → 0, and we would 'lose' a physical VEV. Thus, we must have γ = 1/2, β = 0. Hence, u 2 vanishes on both sides of the degeneration when x → 0, and u 4 survives only on the gauge-theory fixture side. There are no gauge group VEVs supported on the cylinder, as we expected. Also, the parameter u 4 is a square, which is consistent with the a-constraint c This expression can be written, as before, as the product of two global factors: Let us pick the first factor in this expression, and use again the transformation z = t 2 , y = y ′ /2t. We get: where w 2 = x. This is again the Seiberg-Witten curve for the A 1 four-punctured sphere representation of the SU(2) N f = 4 theory, at gauge coupling (4.5).
What about the second factor in (4.9)? One arrives at a result similar to (4.10), but with u 2 and w traded for −u 2 and −w, respectively. So, this also represents the SU(2) gauge theory, with gauge coupling controlled by a cross ratio q ′ = −1+w 1+w . Notice that q ′ = 1/q, and since the points at q and 1/q are related by S-duality, both factors in (4.9) represent the gauge theory at the same point in gauge-coupling moduli space. Again, they differ only by the choice of sign in u 2 , which is left unfixed by the a-constraint c

How it arises
Let us study a sphere with one minimal untwisted and two minimal twisted punctures bubbling off a plane: . . .

[2N + 1] empty
Here the maximal untwisted puncture, [1 2N ], at the right end of the cylinder has SU(2N ) flavour symmetry, but only an SU(N )×SU(N ) subgroup is gauged. Counting hypers reveals that the sphere to the left must be empty. Both SU(N ) factors become weakly coupled when the cylinder degenerates. The second SU(N ) factor is underlined to indicate, as we will see, that there is a specific degeneration of the empty sphere that takes the SU(N ) gauge coupling to zero, but does not decouple the non-underlined SU(N ) factor.

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Let us look at the degeneration where two [2N + 1] collide: As one would expect, counting hypers tells us that the empty sphere decomposes into two empty fixtures. The underlined SU(N ) gauge group is identified with the SU(N ) 2 on the cylinder on the left. One can make the SU(N ) 2 gauge group weakly coupled by degenerating either of the two shown cylinders. Degenerating completely the cylinder to the left turns off the SU(N ) 2 gauge coupling, and leaves just the SU(N ) 1 gauge group supported on the the cylinder on the right.
Instead, if we degenerate the cylinder to the right, both SU(N ) 1 and SU(N ) 2 factors decouple, and we are left with an empty four-punctured sphere on the left. Notice that the four-punctured sphere cannot be a conformal SU(N ) gauge theory because it contains no hypers.
Let us look at this degeneration in more detail. We already saw that the collision of two minimal twisted punctures, [2N + 1], yields a mass-deformed [2 N ] puncture with residue (3.23) and an SU(N ) gauge group. As in (3.23), we take the m i (with N i=1 m i = 0) to be the mass deformations of the [2 N ] puncture.
Similarly, we can consider the OPE of the [2 N ] puncture with the massless [2N − 1, 1] puncture, using the prescription of section 3.3.1. The resulting residue can be diagonalized to the form diag(m 1 , m 2 , . . . , m N ; r 1 , r 2 , . . . , r N ), (4.11) with N i=1 r i = 0. The r i are related to the m i , but do not vanish if the m i are set to zero. Since all terms in (4.11) are generically different, this boundary condition must correspond to an irregular version of the [1 2N ] puncture. In this case, we have two independent sl(N ) Lie algebras, with mass deformations m i and r i , embedded in the sl(2N ) global symmetry group of the [1 2N ] puncture. Clearly, the sl(N ) factor with masses m i is identified with the sl(N ) global symmetry group of the [2 N ] puncture. In particular, if we turn off the m i , we are still left with a sl(N ) factor, with Coulomb branch VEVs r i . This is how the SU(N ) × SU(N ) cylinder arises.

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Consider also the degeneration where [2N + 1] and [2N − 1, 1] collide: . . . We leave the discussion of the dependence of the gauge couplings on the cross-sections to section 5, which is an example that covers all the cases of atypical degenerations.

SU(2) × SU(2) cylinder
Let us now look at a sphere with two minimal untwisted and one minimal twisted punctures, bubbling off a plane. This example is similar to the previous one, but it involves an SU(2) × SU(2) cylinder.
Here, the full SU(2) × SU(2) flavour symmetry of the [2N − 3, 1 4 ] puncture is gauged. (For N = 2, the puncture is [1 5 ], and the global symmetry group is enhanced to Sp(2), but only an SU(2) × SU(2) subgroup is gauged.) The sphere to the left contains four free hypers. Degenerating the cylinder decouples both SU(2) factors. As in the previous example, a certain degeneration of the four-punctured sphere, turns off only the underlined SU(2) factor. Such a degeneration is: As before, the underlined SU(2) is identified with the SU(2) on the left-hand cylinder. Notice that each fixture contains two hypers charged under one of the two SU(2) gauge group factors. The hypers in the left fixture are charged under SU(2) 2 . When the left cylinder decouples, the two hypers in the left fixture become free. If we instead degenerate the right cylinder, both SU(2) gauge groups decouple, and we get a 4-punctured sphere with four free hypers. This is a mass-deformed form of the [2N − 3, 1 4 ] puncture. Here r depends on t, m and n, but does not vanish if these parameters are set to zero. Thus, r and t parametrize the Coulomb branches of the SU(2) 1 and SU(2) 2 gauge groups, respectively.
The second degeneration is: Here the middle fixture contains the four hypers, while the left fixture is empty. Degenerating the left cylinder does not completely decouple either of the SU(2) factors, but degenerating the left cylinder decouples both. In this latter degeneration, the resulting four-punctured sphere again contains four free hypers.
We As we did for the case of the SU(N ) × SU(N ) cylinder, we leave the discussion of the dependence of the gauge couplings on the cross-sections to section 5.

D-shaped quivers
Consider the extended Dynkin diagrams for the simply-laced Lie algebras, given in figure 1, where we have indicated the Dynkin label of each node. It is well known that one obtains a conformally-invariant quiver gauge theory by assigning an SU(l i N ) gauge theory to the i-th node (whose Dynkin label is l i ), and a hypermultiplet, in the bi-fundamental, to each link. It has not, however, been known whether all of these affine quiver gauge theories can be realized as compactifications of the (2,0) theory. The realization of the affine A n quivers is well-known: compactify the A N −1 theory on a torus with n simple punctures. In this section, we present the analogous six-dimensional realization of the affine D n quivers in the twisted A 2N −1 theory. This was first found by Kapustin [8] using a chain of string dualities. We will first present our construction using twisted punctures, and then compare it with Kapustin's.
On the other hand, it is also known that any quiver gauge theory with SU gauge groups which is semiclassically conformal has its gauge groups arranged in the form of a non-affine Dynkin diagram, with SU(N i ) gauge groups on the i-th node and bifundamentals associated to the edges, together with some fundamental flavours at each of the nodes. The realization of the non-affine A n -shaped quivers is known: compactify the A N −1 theory on an untwisted sphere with two regular punctures and a number of simple punctures. At the end of this section, we show how an arbitrary non-affine D n -shaped quiver can be analogously realized in the twisted A 2N −1 theory.

Affine D n -shaped quivers
The affine D n -shaped quiver arises from the compactification of the Here we show only one of the two ends of the affine D n quiver, since the other end is identical. The SU(2N ) cylinders here represent the nodes with a label "2" in the affine D-series Dynkin diagram in the figure above, and the bifundamental fixtures represents the links. The non-trivial piece is the nodes with a "1" at each end of the Dynkin diagram, which correspond to SU(N ) gauge groups. In the sphere above, this piece is represented by the 5-punctured sphere at the left end of the figure. We have deliberately not degenerated the 5-punctured sphere at the end of quiver, since the punctures there are in combinations that lead to atypical degenerations, as studied in the previous section. Let us then examine the degenerations of this 5-punctured sphere in detail.  (2)). The Lagrangian field theory arises only in an "atypical" degeneration (in the nomenclature of the previous section).

Degenerations of the 5-punctured sphere
Degeneration A. The only degeneration that can be understood in the usual, nonatypical sense is: The SU(2) and the SU(N ) gauge groups shown here become weakly coupled as their respective cylinders degenerate. The representations for the matter above are those for the product SU(2) × SU(N ) × SU(2N ).
The next five degenerations are all atypical.
Degeneration B. In this degeneration all of the dynamics is supported on the middle fixture, which is a gauge theory fixture: Here, the underlined SU(N ) is identified with the SU(N ) cylinder on the right hand side. It goes to zero gauge coupling when either cylinder pinches off.
Degeneration D. In this degeneration, the two SU(N ) gauge couplings become equal when the cylinder on the right pinches off.

The moduli space of coupling constants vs. the complex structure moduli space
Let us locate the punctures at The ring of meromorphic functions onM 0,5 consists of all rational functions of the independent cross-ratios Then the compactified moduli space of the 5-punctured sphere,M 0,5 is obtained by blowing up CP 1 × CP 1 described by s 1 and s 2 at 3 points; the result is a del Pezzo surface, dP 4 . The boundary divisor consists of 10 rational curves, D ij , 1 ≤ i < j ≤ 5, corresponding to the locus where the points z i and z j collide. Each D ij has self-intersection number = −1, and intersects precisely three others D ij ∩ D kl = +1 for i, j, k, l all distinct. (5.3) The moduli space of the coupling constants, however, is notM 0,5 . Instead, it is a 4-sheeted branched cover X 5 →M 0,5 , branched over the compactification divisor. We will be more precise about the nature of the ramification, below, but X 5 is most effectively described as being the rational surface whose ring of meromorphic functions consists of all rational functions of y, w where The UV gauge couplings of our two decoupled gauge theories are where q = 0, ∞ correspond to a weakly-coupled SU(N ) gauge group and q = 1 is the point where the dual SU(2) gauge group is weakly-coupled.
There is a natural action of the dihedral group, D 4 , on our family of gauge theories, generated by α :  While they are easy to compute from (5.2), (5.4) and (5.5), we indicate, in table 1, the number of sheets over a generic point on the divisor and the behaviour of the gauge couplings (5.5) on the pre-images of each of the components of the compactification divisor. E.g., on one of the pre-images of D 34 , q 1 ≡ 1, while q 2 varies. On the other pre-image, q 1 varies, while q 2 ≡ 1. From these, we easily see that the behaviour of the gauge theory, at each of the degenerations discussed in the previous subsection, is as we claimed. For instance, on D 13 ∩ D 24 (or D 14 ∩ D 23 ), we have y = 0, w = ∞ (y = ∞, w = 0) and hence q 1 = q 2 = −1.

Comparison to Kapustin's work
In [8], Kapustin realized the affine D-shaped quiver in Type IIA string theory. As always, consider 2N D4-branes extending along directions 01236, and k NS5-branes extending along directions 012345. Furthermore, introduce a suitable orbifold whose action includes Divisor # sheets (q 1 , q 2 ) (1, 1)  This particular orbifold is known to be magnetically charged under the B-field, like an NS5-brane. The M-theory lift of the configuration is then given by 2N M5-branes wrapped on a torus parametrized by z, together with the M-theory orientifold action z → −z, which has four fixed points. Each of the two orbifold planes lifts to a pair of fixed points, each pair having an M5-brane on top of it. We also have k M5-branes intersecting the torus. We can move the two M5-branes away from the fixed points, and the final configuration becomes 2N M5-branes on a torus with the orientifold action z → −z, plus k + 2 M5-branes. We can take the decoupling limit. Each of the M-theory orientifold fixed points becomes a twisted simple puncture of type [2N + 1]; the torus divided by z → −z is a sphere; and the intersection with an M5-brane is a untwisted simple puncture of type [2N − 1, 1]. Thus we have the 6D theory of type A 2N −1 on a sphere with four twisted punctures of type [2N + 1] and k + 2 untwisted simple punctures of type [2N − 1, 1], which reproduces our previous analysis.
We saw above that the degeneration limit where an untwisted simple puncture collides with a twisted simple puncture corresponds to the point where the two gauge couplings of an SU(N ) × SU(N ) gauge group become equal. In the M-theory construction, this corresponds to the fact that to take the IIA limit, two of the fixed points need to be paired, with an M5-brane on top of them.
The cylinder connecting the two full punctures has one side spontaneously broken to SU(n 1 ) × SU(n 1 ), and the other to SU(n 1 + n 2 ). So, in the 4D limit, it supports an SU(n 1 ) × SU(n 2 ) gauge group. The combined system hence realizes the non-affine Dshaped quiver (5.8).

SU(4) and Sp(2) gauge theories
As an application of the A 3 twisted theory, we study below S-duality of the SU(4) and Sp (2) superconformal gauge theories with matter in all allowed combinations of the fundamental and antisymmetric representations. The full tables of twisted and untwisted punctures, cylinders and fixtures of the A 3 theory are shown in appendix A.1.

4(6)
Finally, we can take both fixtures from the twisted sector.

6(4)
This classic example of Argyres-Seiberg duality is realized in the untwisted sector of the A 3 theory.
Degeneration A.
• The new SCFT with ∆(u) = 4 is obtained from the A 3 theory on a sphere with an untwisted puncture of type [2, 1 2 ] and two twisted punctures of type [2 2 , 1]. This realization comes with a free half-hypermultiplet in the fundamental of the SU(2) flavour symmetry. This is the example we studied in section 6.2.3.

On a new rank-1 SCFT with ∆(u) = 3
To obtain the new SCFT with ∆(u) = 3, we need to extend our analysis to the twisted A 2n theory. The Z 2 twist of the A 2n theory is particularly subtle, as emphasized in [13]. Hence, we prefer to postpone a systematic analysis of the twisted A 2n theory. Still, it is possible to show how to obtain the missing new SCFT from a 6D construction.
In [11], the new theory with ∆(u) = 3 is introduced in the following way. Consider the SU(3) gauge theory with one hyper in the fundamental and one hyper in the symmetric tensor representation. The S-dual theory is an SU(2) gauging of the new SCFT, coupled to n half-hypermultiplets in the doublet. The field-theory arguments in [12] constrain n to be 0 or 2, and require the flavour symmetry h of the SCFT to satisfy k h = (8 − n)/I, where I is the index of the embedding of su(2) in h. Now, the tensor product of two fundamentals of SU(3) decomposes as the direct sum of a fundamental plus a symmetric representation. The tensor product can in turn be obtained from the bifundamental of a product group SU(3) 1 ×SU(3) 2 , by taking a diagonal subgroup SU(3) diag , such that SU(3) diag is embedded in SU(3) 1 in the standard way, but embedded in SU(3) 2 with the action of complex conjugation, i.e., the nontrivial outer automorphism.
So, consider the A 2 theory on a fixture with a simple puncture and two full punctures, which by itself simply gives rise to a bifundamental. Then, the SU(3) gauge theory with matter in the 1(3) + 1(6) can be realized by connecting the full punctures in the fixture by a cylinder with a Z 2 twist line looping around it. In other words, we have the A 2 theory on a torus with one simple puncture and a Z 2 twist loop. See the left side of the figure below. SU(3) SU (2) In the S-dual frame, shown on the right side of the figure, we have a fixture with an untwisted simple puncture and two twisted full punctures (denoted by ), and the full punctures are connected by a cylinder with a Z 2 -twist line along the cylinder. Clearly, this gives a weakly-coupled SU(2) gauge field coupled to the fixture. Also, the flavour symmetry group of the fixture must contain the explicit SU(2) 2 × U(1) as a subgroup. Thus, based on our findings, the fixture may be: • an interacting SCFT with flavour symmetry group H ⊃ SU(2) 2 × U(1) (if n = 0), or • an interacting SCFT with flavour symmetry group H ⊃ SU(2) 2 × U(1), plus free hypers in the (2, 1) + (1, 2) of SU(2) 2 and neutral under U(1) (if n = 2).
To see which of these two possibilities is the right one, recall [18] that when the Riemann surface has two twisted (or untwisted) full punctures with flavour symmetry G, the holomorphic moment maps µ 1 , µ 2 of the two G-actions on the Higgs branch must be equal, tr µ 1 2 = tr µ 2 2 (7.1) Let us see what happens if n = 2. In this case, the Higgs branch is X × H 1 × H 2 , where X is the Higgs branch of the interacting SCFT with an action of SU(2) 1 × SU(2) 2 , and H 1,2 is the Higgs branch for the SU(2) 1,2 free hypers, respectively. Then we have tr µ i 2 = tr µ X,i 2 + tr µ 2 H i (7.2) for i = 1, 2. However, tr µ 1 2 depends on the position on H 1 , but not on the position on H 2 , while for tr µ 2 2 the opposite is true. Hence, n = 2 does not satisfy the condition (7.1). Then, we conclude that n = 0, and so the A 2 fixture with one untwisted simple puncture and two twisted full punctures contains just the new interacting rank-1 SCFT with ∆(u) = 3.